Earth and space observations meet complex algebras: from complex to octonions for multivariate autoregressive time series analysis
Abstract
Many datasets arise as multivariate time series observed at irregular time intervals, particularly in Earth observation and astronomical measurements. Classical time-series models assume that time is discrete and observations occur at equally spaced intervals, limiting their ability to capture dynamics when observation gaps vary. Existing approaches to irregular sampling rely on continuous-time formulations, which assume that observation intervals are sufficiently small. We address this limitation by introducing a discrete-time framework that accommodates irregular observation gaps while preserving multivariate dependence structures. We introduce a framework for the analysis of irregularly observed multivariate time series based on hypercomplex autoregressive processes. The framework embeds multivariate observations into a hypercomplex algebraic structure, allowing multiple variables and their interactions to be represented within a single mathematical entity while incorporating irregular observation gaps. The resulting process admits a structured matrix representation, which enables the model to be expressed as a state-space system. This formulation provides an estimation methodology in which model parameters can be inferred using Kalman filtering techniques. The state-space representation allows recursive estimation and prediction in the presence of irregular sampling intervals. The performance of the methodology is illustrated through data and applications to remote sensing and astronomical datasets, where observations occur at nonuniform time intervals. The results demonstrate that the approach captures dynamics and cross-variable interactions difficult to model using traditional time-series techniques. The proposed framework provides a methodology for analyzing irregularly sampled multivariate time series and opens possibilities for modeling observational data across disciplines.
Related
- Multivariate Continuous-Time Autoregressive Moving Average Processes for Astronomical Multiband Time Series
- Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators
- Reduced Solutions of Toda Systems for Arbitrary Complex Simple Lie Algebras
- Octonions, Albert vectors and the group 2 E6(F)
- The Dirac equation in (split-)octonions: origins, variants, and modern context
BibTeX
@misc{eyheramendy2026earth,
title = {{Earth and space observations meet complex algebras: from complex to octonions for multivariate autoregressive time series analysis}},
author = {Susana Eyheramendy and Felipe Elorrieta and Wilfredo Palma and Javier Lopatin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18794},
eprint = {2609.18794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18794},
}
RIS
TY - UNPB
TI - Earth and space observations meet complex algebras: from complex to octonions for multivariate autoregressive time series analysis
AU - Susana Eyheramendy
AU - Felipe Elorrieta
AU - Wilfredo Palma
AU - Javier Lopatin
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18794
UR - https://arxiv.org/abs/2609.18794
AB - Many datasets arise as multivariate time series observed at irregular time intervals, particularly in Earth observation and astronomical measurements. Classical time-series models assume that time is discrete and observations occur at equally spaced intervals, limiting their ability to capture dynamics when observation gaps vary. Existing approaches to irregular sampling rely on continuous-time formulations, which assume that observation intervals are sufficiently small. We address this limitation by introducing a discrete-time framework that accommodates irregular observation gaps while preserving multivariate dependence structures. We introduce a framework for the analysis of irregularly observed multivariate time series based on hypercomplex autoregressive processes. The framework embeds multivariate observations into a hypercomplex algebraic structure, allowing multiple variables and their interactions to be represented within a single mathematical entity while incorporating irregular observation gaps. The resulting process admits a structured matrix representation, which enables the model to be expressed as a state-space system. This formulation provides an estimation methodology in which model parameters can be inferred using Kalman filtering techniques. The state-space representation allows recursive estimation and prediction in the presence of irregular sampling intervals. The performance of the methodology is illustrated through data and applications to remote sensing and astronomical datasets, where observations occur at nonuniform time intervals. The results demonstrate that the approach captures dynamics and cross-variable interactions difficult to model using traditional time-series techniques. The proposed framework provides a methodology for analyzing irregularly sampled multivariate time series and opens possibilities for modeling observational data across disciplines.
ER -